What are the standard problems for CNNs and LSTMs?
It's more domain- or task-specific. There is no obvious baseline anymore because these models and this field has evolved into too large of an ecosystem. Nonetheless, I'll list a couple of notorious examples below.

Image classification:  


MNIST
CIFAR  
ImageNet


Detection/segmentation:  


PascalVOC  
COCO  
CityScapes  


Pose estimation:  


MPII  
LEEDS


Text classification:  


IMDB  
yelp  


Question answering:  


SQuAD  


Translation:  


WMT
IWSLT  


This is just a taste. There are tons more both in each category and the number of categories, a good source is the Papers with Code website.  

Therefore, there is no single standard problem, given that there are too many that all in one shape or form use CNNs or RNNs (or others).